Due to convexity of \(f\) , we have \(D_f(P\Vert Q)\ge f(1) = 0\) . If, furthermore, \(f\)  is strictly convex at 1, then equality holds if and only \(P=Q\) . Popular examples of \(f\) -divergences include \(f(t) = t\log t\)  corresponding to KL divergence, \(f(t) = |t-1|\)  corresponding to total variation distance, and \(f(t) = t^2-1\)  corresponding to \(\chi ^2\) -divergence.
In this paper, we mostly concern with an important sub-family of \(f\) -divergences
associated with \(f_\gamma (t) = \max \lbrace t-\gamma , 0\rbrace \) 
for a parameter \(\gamma \ge 1\) .
The corresponding \(f\) -divergence, denoted by \({\mathsf {E}}_\gamma (P\Vert Q)\) , is called \({\mathsf {E}}_\gamma \) -divergence (or sometimes hockey-stick divergence [1]}) and is explicitly defined in (REF ). It appeared in [2]} for proving converse channel coding results and also used in [3]}, [4]}, [5]}, [6]} for characterizing privacy guarantees of iterative algorithms in terms of other variants of DP.
